The automorphisms group and the classification of gradings of finite dimensional associative algebras

Let $A$ be an $n$-dimensional algebra over a field $k$ and $a(A)$ its quantum symmetry semigroup. We prove that the automorphisms group ${\rm Aut}_{\rm Alg} (A)$of$A$isisomorphictothegroup$U \bigl( G(a (A)^{\rm o} ) \bigl)$ of all invertible group-like elements of the finite dual $a(A{)}^{\mathrm{o}}$. For a group $G$, all $G$-gradings on $A$ are explicitly described and classified: the set of isomorphisms classes of all $G$-gradings on $A$ is in bijection with the quotient set ${\mathrm{H}\mathrm{o}\mathrm{m}}_{\mathrm{B}\mathrm{i}\mathrm{A}\mathrm{l}\mathrm{g}}(a(A),k[G])\mathrm{/}\approx$ of all bialgebra maps $a(A)\to k[G]$, via the equivalence relation implemented by the conjugation with an invertible group-like element of $a (A)^{\rm o}$.